Finite versus uncountable convex lattices from point configurations

Abstract

We study the smallest convex lattice generated by a finite set of points. To analyze this structure, we introduce the notion of a point configuration, defined via the relative lattice. Under a suitable completeness condition, this lattice becomes a combinatorial counterpart of the convex lattice and is therefore easier to handle. We investigate the enumeration of these structures and prove that, while the number of relative lattices is always finite, the number of convex lattices is uncountable for $n \geq 6$.